Homotopy and algebraicK-theory
Barry H. Dayton · Pacific Journal of Mathematics · 1972
A notion of homotopy is described on a category of rings.This is used to induce a notion of equivalence on the categories of projective modules and to construct a iΓ-theory exact sequence.The topological i£-theory exact sequence is then obtained from the algebraic K o , K x sequence.1* Homotopy* In this section we describe the homotopy notion and the notion of equivalence it induces on the categories of projective modules.A cartesian square of rings is a commutative diagram of rings (*) Jλ, \ > Λ Qwhere A = {(a l9 α 2 ) e A γ x A 2 \fι(a^ = / 2 (α 2 )} and h iy h 2 are restrictions of the coordinate projections.We will further assume that f λ is surjective.If S%Γ is a category of rings and F: Sf -* 3ίΓ is a functor we call F cartesian square preserving if the functor applied to a cartesian square gives a cartesian square.DEFINITION 1.1.Let SΓ be a category of rings.A homotopy theory Sίf for J^ is an ordered quadruple (I, c Q , c l9 π) where I is a cartesian square preserving functor and c 0 , ^: /-> 1^, π: 1^ ->I are natural transformations such that c o (A)π(A) = 1,1 = c ι {A)π(A) for A e